In a digital communication system comprising a transmitter and a receiver, a digital signal to be transmitted—usually a succession of symbols—is converted, before transmission, into a continuous-time analog signal, which is then transmitted through a physical propagation medium, air or any other physical propagation environment. When the signal is received by the receiver, it is then processed and converted into digital form by means of appropriate sampling, which is typically carried out at a frequency fe that should be synchronous to symbol emission frequency fs. Unfortunately, the clocks situated in the oscillating circuits equipping the transmitter and the receiver are never synchronous and it is then necessary to compensate for any frequency drift between these clocks, in order to be able to correctly process the received signal and to extract emitted symbols.
Such frequency shift affecting oscillators at transmission and reception generates a parasitic phase shift in the output signal of the complex demodulator located in the receiver. Other factors contribute to accentuate this parasitic phase shift. First, there is the time needed by digital signals to flow through a propagation medium. Secondly, any movement of the transmitter relative to the receiver generates Doppler beat and tends to introduce further disruptive phase shift.
Referring to a baseband model, observations Yk at the output of the complex demodulator located in the receiver can be expressed by the following formula:Yk=akeiξk+ηk 
where ak corresponds to emitted symbols, ξk is the parasitic phase shift and ηK is additional noise.
Techniques are already known—based on phase estimator circuits—to estimate this parasitic phase shift ξK and correct it.
The most sophisticated phase estimators, which process simultaneously whole sets of received observations, are based on extremely cumbersome digital processing: particulate filtering, random-walk methods with Markov chain, etc. In practice, implementing such techniques proves to be unrealizable due to the considerable computing power required.
For this reason, the complexity of phase estimators is opposed to the simple implementation of phase-locked loops that sequentially process received observations one after the other instead of processing whole sets of received observations. Typically, a phase locked loop (PLL) is based on an iterative digital algorithm for estimating a phase estimate. Whereas traditionally, phase-locked loops were carried out by means of analog circuits, now such processing is purely digital. It should be noted that this digital processing depends closely on the type of modulation considered.
As an example, let us consider the case of a binary phase shift keying (BPSK) modulation. In such BPSK modulation, transmitted symbols ak are equal to −1 or +1. Because of the previously mentioned parasitic phase shift, one doesn't obtain −1 nor +1 at the output of the complex demodulator, but these values modified by a phase shift. A well-known PLL for correcting such phase shift is the one known as Costa's loop that relies on the use of a gradient algorithm, associated with a cost function J given by the following formula:J(φ)=E(|y2ke−i2φ−1|2)
where E is the Expectation operator.
Applying the gradient algorithm to variable φ makes it possible to make the algorithm converge towards a phase estimate:φk=φk−1−γ∂J(φ)/∂φ|φ=φk−1 
A Costa's loop is finally obtained by removing the expectation: it is the stochastic gradient algorithm minimizing cost function J.
Other formulas are known for other types of modulation and in particular squaring modulation, also known as four-state Quadrature Amplitude Modulation or 4-QAM. Generally, whatever the type of modulation employed, phase-locked loops are built according to a general formula of the type:φk=φk−1−γF(Yk,φk−1)
where F is a function depending closely on the considered modulation.
Typically, as can be seen from the preceding formula, all loops consist in calculating a phase φk according to the preceding element φk−1 and a function F of both elements Yk and φk−1. It should be noted that, in this formula, according to the type of phase shift to be corrected, sophisticated correction of the parameter γ can be used, and in particular a corrective second-order filter (proportional integral), or even a higher order filter, could be used.
All known phase-locked loops—conventionally adopting the known analog model—present the same limitation. The evaluation of phase φK is primarily based on the preceding phase value φk−1 and on a function of one or more previous observations. Hence, an imperfect estimation of the phase and, consequently, correction thereof.
It is advisable to improve the phase locked loop (PLL) model in order to increase precision of the estimate and effectiveness of the correction process.